scholarly journals Free Lie algebras as modules for symmetric groups

1999 ◽  
Vol 67 (2) ◽  
pp. 143-156 ◽  
Author(s):  
R. M. Bryant ◽  
L. G. Kovács ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebras ◽  
Symmetric Groups ◽  
Free Lie Algebra ◽  
Prime Characteristic ◽  
Specht Modules ◽  
Generating Set ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Free Lie Algebras ◽  
Direct Decompositions

AbstractLet r be a positive integer, F a field of odd prime characteristic p, and L the free Lie algebra of rank r over F. Consider L a module for the symmetric group , of all permutations of a free generating set of L. The homogeneous components Ln of L are finite dimensional submodules, and L is their direct sum. For p ≤ r ≤ 2p, the main results of this paper identify the non-porojective indecomposable direct summands of the Ln as Specht modules or dual Specht modules corresponding to certain partitions. For the case r = p, the multiplicities of these indecomposables in the direct decompositions of the Ln are also determined, as are the multiplicities of the projective indecomposables. (Corresponding results for p = 2 have been obtained elsewhere.)

scholarly journals THE DECOMPOSITION OF LIE POWERS

2006 ◽  
Vol 93 (1) ◽  
pp. 175-196 ◽  
Author(s):  
R. M. BRYANT ◽  
M. SCHOCKER
Keyword(s):  
Positive Integer ◽  
Free Associative Algebra ◽  
Free Lie Algebra ◽  
Tensor Algebra ◽  
Prime Characteristic ◽  
Characteristic P ◽  
Descent Algebra ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Solomon Descent Algebra

Let $G$ be a group, $F$ a field of prime characteristic $p$ and $V$ a finite-dimensional $FG$-module. Let $L(V)$ denote the free Lie algebra on $V$ regarded as an $FG$-submodule of the free associative algebra (or tensor algebra) $T(V)$. For each positive integer $r$, let $L^r (V)$ and $T^r (V)$ be the $r$th homogeneous components of $L(V)$ and $T(V)$, respectively. Here $L^r (V)$ is called the $r$th Lie power of $V$. Our main result is that there are submodules $B_1$, $B_2$, ... of $L(V)$ such that, for all $r$, $B_r$ is a direct summand of $T^r(V)$ and, whenever $m \geqslant 0$ and $k$ is not divisible by $p$, the module $L^{p^mk} (V)$ is the direct sum of $L^{p^m} (B_k)$, $L^{p^{m - 1}} (B_{pk})$, ..., $L^1 (B_{p^mk})$. Thus every Lie power is a direct sum of Lie powers of $p$-power degree. The approach builds on an analysis of $T^r (V)$ as a bimodule for $G$ and the Solomon descent algebra.

WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA

2004 ◽  
Vol 15 (10) ◽  
pp. 987-1005 ◽  
Author(s):  
MAHMOUD BENKHALIFA
Keyword(s):  
Exact Sequence ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Integral Domain ◽  
Free Lie Algebra ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Equivalent Class

Let R be a principal and integral domain. We say that two differential graded free Lie algebras over R (free dgl for short) are weakly equivalent if and only if the homologies of their corresponding enveloping universal algebras are isomophic. This paper is devoted to the problem of how we can characterize the weakly equivalent class of a free dgl. Our tool to address this question is the Whitehead exact sequence. We show, under a certain condition, that two R-free dgls are weakly equivalent if and only if their Whitehead sequences are isomorphic.

scholarly journals Soluble Lie algebras having finite-dimensional maximal subalgebras

1974 ◽  
Vol 11 (1) ◽  
pp. 145-156 ◽  
Author(s):  
Ian N. Stewart
Keyword(s):  
Lie Algebras ◽  
Minimal Ideal ◽  
Prime Characteristic ◽  
Maximal Subalgebra ◽  
Maximal Subalgebras ◽  
Infinite Dimensional ◽  
The Core ◽  
Zero Characteristic ◽  
Finite Dimensional ◽  
Split Extensions

Infinite-dimensional soluble Lie algebras can possess maximal subalgebras which are finite-dimensional. We give a fairly complete description of such algebras: over a field of prime characteristic they do not exist; over a field of zero characteristic then, modulo the core of the aforesaid maximal subalgebra, they are split extensions of an abelian minimal ideal by the maximal subalgebra. If the field is algebraically closed, or if the maximal subalgebra is supersoluble, then all finite-dimensional maximal subalgebras are conjugate under the group of automorphisms generated by exponentials of inner derivations by elements of the Fitting radical. An example is given to indicate the differences encountered in the insoluble case, and the nonexistence of group-theoretic analogues is briefly discussed.

The Schreier Technique for Subalgebras of a Free Lie Algebra

1997 ◽  
Vol 49 (3) ◽  
pp. 600-616 ◽  
Author(s):  
Shmuel Rosset ◽  
Alon Wasserman
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Free Group ◽  
Finite Codimension ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Free Lie Algebras

AbstractIn group theory Schreier's technique provides a basis for a subgroup of a free group. In this paper an analogue is developed for free Lie algebras. It hinges on the idea of cutting a Hall set into two parts. Using it, we show that proper subalgebras of finite codimension are not finitely generated and, following M. Hall, that a finitely generated subalgebra is a free factor of a subalgebra of finite codimension.

THE EQUATION [x,u] + [y,v] = 0 IN FREE LIE ALGEBRAS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1165-1187 ◽  
Author(s):  
VLADIMIR REMESLENNIKOV ◽  
RALPH STÖHR
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Solution Space ◽  
Free Generator ◽  
Finite Codimension ◽  
Free Lie Algebra ◽  
Free Lie Algebras ◽  
Free Generators ◽  
Bilinear Equation

We investigate equations of the form [x,u] + [y,v] = 0 over a free Lie algebra L. In the case where u and v are free generators of L, we exhibit two series of solutions, we work out the dimensions of the homogeneous components of the solution space, and we determine its radical. In the general case we show that the results on free generator coefficients are sufficient to obtain the solution space up to finite codimension. As an application we determine the radical of the bilinear equation [x1,x2] + [x3,x4] = 0.

scholarly journals Subalgebras of free restricted Lie algebras

2005 ◽  
Vol 72 (1) ◽  
pp. 147-156 ◽  
Author(s):  
R.M. Bryant ◽  
L.G. Kovács ◽  
Ralph Stöhr
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Main Step ◽  
Free Lie Algebra ◽  
Generating Set ◽  
Restricted Lie Algebra ◽  
Corrected Proof ◽  
Restricted Lie Algebras ◽  
Remarkable Result ◽  
Algebra Over A Field

A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshov's proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshov's argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere.

scholarly journals Lie algebras with S3- or S4-action and generalized Malcev algebras

2009 ◽  
Vol 139 (2) ◽  
pp. 321-357 ◽  
Author(s):  
Alberto Elduque ◽  
Susumu Okubo
Keyword(s):  
Lie Algebras ◽  
Symmetric Groups ◽  
Triple Systems ◽  
Direct Sum ◽  
Irreducible Modules ◽  
Jordan Triple Systems ◽  
The Given

Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied. In the case of S3-symmetry, the Lie algebras are coordinatized by some non-associative systems, which are termed generalized Malcev algebras, as they extend the classical Malcev algebras. These systems are endowed with a binary and a ternary product, and include both the Malcev algebras and the Jordan triple systems.

scholarly journals MULTISTRING VERTICES AND HYPERBOLIC KAC-MOODY ALGEBRAS

1996 ◽  
Vol 11 (03) ◽  
pp. 429-514 ◽  
Author(s):  
R.W. GEBERT ◽  
H. NICOLAI ◽  
P.C. WEST
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Complete Information ◽  
Liouville Theory ◽  
Root Lattice ◽  
Root Space ◽  
Respective Group ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras

Multistring vertices and the overlap identities which they satisfy are exploited to understand properties of hyperbolic Kac-Moody algebras, in particular E10. Since any such algebra can be embedded in the larger Lie algebra of physical states of an associated completely compactified subcritical bosonic string, one can in principle determine the root spaces by analyzing which (positive norm) physical states decouple from the N-string vertex. Consequently, the Lie algebra of physical states decomposes into a direct sum of the hyperbolic algebra and the space of decoupled states. Both these spaces contain transversal and longitudinal states. Longitudinal decoupling holds more generally, and may also be valid for uncompactified strings, with possible consequences for Liouville theory; the identification of the decoupled states simply amounts to finding the zeroes of certain “decoupling polynomials.” This is not the case for transversal decoupling, which crucially depends on special properties of the root lattice, as we explicitly demonstrate for a nontrivial root space of E10· Because the N-vertices of the compactified string contain the complete information about decoupling, all the properties of the hyperbolic algebra are encoded into them. In view of the integer grading of hyperbolic algebras such as E10 by the level, these algebras can be interpreted as interacting strings moving on the respective group manifolds associated with the underlying finite-dimensional Lie algebras.

Automorphisms of the completion of relatively free Lie algebras

2018 ◽  
Vol 28 (06) ◽  
pp. 1091-1100
Author(s):  
C. E. Kofinas
Keyword(s):  
Automorphism Group ◽  
Power Series ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Free Lie Algebra ◽  
Central Series ◽  
Dense Subgroup ◽  
Free Lie Algebras ◽  
Finite Set ◽  
Formal Power

Let [Formula: see text] be a relatively free Lie algebra of finite rank [Formula: see text], with [Formula: see text], [Formula: see text] be the completion of [Formula: see text] with respect to the topology defined by the lower central series [Formula: see text] of [Formula: see text] and [Formula: see text], with [Formula: see text]. We prove that, with respect to the formal power series topology, the automorphism group [Formula: see text] of [Formula: see text] is dense in the automorphism group [Formula: see text] of [Formula: see text] if and only if [Formula: see text] is nilpotent. Furthermore, we show that there exists a dense subgroup of [Formula: see text] generated by [Formula: see text] and a finite set of IA-automorphisms if and only if [Formula: see text] is generated by [Formula: see text] and a finite set of IA-automorphisms independent upon [Formula: see text] for all [Formula: see text]. We apply our study to several varieties of Lie algebras.

scholarly journals ON THE DIMENSION OF PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS

2013 ◽  
Vol 23 (01) ◽  
pp. 205-213 ◽  
Author(s):  
NIL MANSUROǦLU ◽  
RALPH STÖHR
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Rank ◽  
Homogeneous Component ◽  
Free Lie Algebra ◽  
Dimension Function ◽  
Free Lie Algebras ◽  
Explicit Formulae ◽  
Characteristic 2

Let L be a free Lie algebra of finite rank over a field K and let Ln denote the degree n homogeneous component of L. Formulae for the dimension of the subspaces [Lm, Ln] for all m and n were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form [Lm, Ln, Lk] = [[Lm, Ln], Lk]. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field K. For example, the dimension of [L2, L2, L1] over fields of characteristic 2 is different from the dimension over fields of characteristic other than 2. Our main results are formulae for the dimension of [Lm, Ln, Lk]. Under certain conditions on m, n and k they lead to explicit formulae that do not depend on the characteristic of K, and express the dimension of [Lm, Ln, Lk] in terms of Witt's dimension function.

Sign in / Sign up

Export Citation Format

Share Document